Infinite staircases for generalized equivariant embedding problems

Determine whether the generalized equivariant embedding sets \(\mathcal E^{\mathrm{eq}}_{d;p,q}\), defined for positive integers \(d\) and coprime integers \(0<q\leq p\) by the existence of a \(\Gamma_{d;p,q}\)-equivariant symplectic embedding \(E(dp^2\beta,dp^2\alpha)\hookrightarrow B^4(dp^2)\), exhibit infinite staircase structures.

Background

The paper studies symplectic embedding problems associated with Katok’s examples and identifies the corresponding $Z_2$- and $Z_4$-equivariant ellipsoid embedding problems. In the cases treated, the parameter sets determining when the equivariant embeddings exist contain infinite staircases governed by explicit recursions.

The authors define a broader family of equivariant embedding problems indexed by $(d;p,q)$ and ask whether the staircase phenomenon persists throughout this family. This would extend the results beyond the $(2;1,1)$ and $(1;2,1)$ cases considered in the paper.